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zVCbP^2ot`-v#Loua41co6#BaVu5B!hbs-6Vi%(u@e=AF227QY|1=+`mB?iOi+938V z_E~e`oonk|o+$E9Yla}A_lnf#Qs>4)*IFmbS?d;NX)t1-MVMOA2@Quojb_*=hvneteWL&(L4^So-T75vkWK!0w)(YD<>QU Kg@lqM-2VXL=S7JC diff --git a/cviceni.tex b/cviceni.tex index 00b9621..b6e6ebb 100644 --- a/cviceni.tex +++ b/cviceni.tex @@ -32,5 +32,6 @@ \include{cviceni1} \include{cviceni2} +\include{cviceni3} \end{document} diff --git a/cviceni3.tex b/cviceni3.tex new file mode 100644 index 0000000..2b303b1 --- /dev/null +++ b/cviceni3.tex @@ -0,0 +1,53 @@ +\section{Cvičení 3} + +Na tomto cvičení se budeme věnovat řešení kvazilineárních PDR. + +\begin{example} + Řešte úlohu $\pdv{u}{t} + b \pdv{u}{x} = u^2$ s počáteční podmínkou $u(t_0, x) = u_0(x)$. +\end{example} + +\begin{proof} + Uvažujme homogenní rovnici $\pdv{w}{t} + b\pdv{w}{x} + z^2 \pdv{w}{z}$ a k ní příslušný charakteristický systém. $\odv{t}{s} = 1, \odv{x}{s} = b, \odv{z}{s} = z^2$. Potom jednoduše vidíme, že $\odv*{(x - bt)}{s} = 0$ a zároveň $\odv*{(\frac{1}{z} + t)}{s} = 0$. + Tedy máme charakteristiky $\psi_1(t, x, z) = x - bt, \psi_2(t, x, z) = \frac{1}{z} + t$. Řešením původní homogenní rovnice je jejich kombinace, tedy $\psi(t, x, z) = F(\psi_1(t, x, z), \psi_2(t, x, z))$. Aplikací počáteční podmínky dostáváme $\psi(t_0, x, u_0(x)) = 0$, tedy hledáme $F(u, v)$ tak, aby $F(x - bt_0, \frac{1}{u_0(x)} + t_0) = 0$. + Takovou funkcí je $F(u, v) = \frac{1}{u_0(u + bt_0)} + t_0 - v$. Zpětným dosazením dostáváme $\psi(t, x, z) = \frac{1}{u_0(x - b(t - t_0))} + t_0 - \frac{1}{z} + t = 0$. Z toho již můžeme vyjádřit $z = \frac{1}{\frac{1}{u_0(x - b(t - t_0))} - t + t_0} = \frac{u_0(x - b(t - t_0))}{1 - (t - t_0)u_0(x - b(t - t_0))}$. +\end{proof} + +\begin{example} + Řešte úlohu $\pdv{u}{t} + x\pdv{u}{x} = -tu$ s počáteční podmínkou $u(0, x) = \sin x$. +\end{example} + +\begin{proof} + Uvažujeme rovnici $\pdv{w}{t} + x\pdv{w}{x} - tz\pdv{w}{z} = 0$. K ní máme soustavu ODR $\odv{t}{s} = 1, \odv{x}{s} = x, \odv{z}{s} = -tz$. Odtud máme $\odv*{(t - \log x)}{t} = 0$ a $\odv*{(\frac{1}{2}t^2 + \log z)}t = 0$. Tedy máme $\psi_1(t, x, z) = t - \log |x|$ a $\psi_2(t, x, z) = \frac{1}{2}t^2 + \log |z|$. Dle předchozího příkladu hledáme řešení ve tvaru $F(\psi_1(t, x, z), \psi_2(t, x, z))$, a tedy takové, že $0 = F(\psi_1(0, x, \sin x), \psi_2(0, x, \sin x)) = F(- \log |x|, \log |\sin x|)$. Uvažujme pouze množinu kde $x > 0, \sin x > 0$. Víme, že $e^{\log \sin x} - \sin e^{\log x} = \sin x - \sin x = 0$, tedy můžeme psát $e^{\psi_2} - \sin e^{-\psi_1} = 0$. Z toho dostáváme $e^{\frac{1}{2}t^2 + \log z} - \sin e^{\log x - t} = 0$. Zbývá už jen vyjádřit $z = e^{-\frac{1}{2}t^2} \sin (xe^{-t})$. + + Alternativní postup: Nalezneme řešení charakteristického systému. Dostaneme $t(s) = s + c_1$, $x(s) = c_2 e^s$, $z'(s) = -(s + c_1)z(s)$, tedy $\log z(s) = - \frac{1}{2}s^2 - c_1 s + c_2$. Z toho dostáváme $z(s) = c_3 e^{-\frac{1}{2}s^2 - c_1 s}$. Fix $t_0, x_0, z_0$. Najdeme hodnoty konstant takové, aby tento stav nastal v čase $s = 0$. Máme $c_1 = t_0$, $c_2 = x_0$, $c_3 = z_0$. Kdy je $t(s)$ nulové? V čase $s = -t_0$. Potom $x(-t_0) = x_0 e^{-t_0}$ a $z(-t_0) = z_0e^{-\frac{1}{2}t_0^2 + t_0^2} = z_0e^{\frac{1}{2}t_0^2}$. + Z počáteční podmínky dostáváme $w(0, x, z) = z - \sin x$, a tedy $w(t, x, z) = z(t) - \sin(x(t)) = ze^{\frac{1}{2}t^2} - \sin(xe^{-t}) = 0$, tedy $z = e^{-\frac{1}{2}t^2} \sin(xe^{-t})$. +\end{proof} + +\begin{example} + Řešte $x\pdv{u}{x} + \pdv{u}{y} = y$ s podmínkou $u(x, 0) = x^2$. +\end{example} + +\begin{proof} + Máme homogenní rovnici $x\pdv{w}{x} + \pdv{w}{y} + y\pdv{w}{z} = 0$. Charakteristický systém je $\odv{x}{s} = x, \odv{y}{s} = 1, \odv{z}{s} = y$. Platí $\odv*{(\log x - y)}s = 0$ a $\odv{(\frac{1}{2}y^2 - z)} = 0$. Máme řešení $F(\log x - y, \frac{1}{2}y^2 - z)$. Dosadíme počáteční podmínku a máme $0 = F(\log x, - x^2) = 0$. Tomu odpovídá například funkce $F(u, v) = e^{2u} + v$. Zpětným dosazením získáme $0 = e^{2\log x - 2y} + \frac{1}{2}y^2 - z$, tedy $z = x^2e^{-2y} + \frac{1}{2}y^2$. +\end{proof} + +\begin{example} + Řešte úlohu $2\pdv{u}{x} + 5\pdv{u}{y} + 6u = 0$, s počáteční podmínkou $u(x, 0) = x\cos x$. +\end{example} + +\begin{proof} + Uvažujme homogenní rovnici $2\pdv{w}{x} + 5\pdv{w}{y} - 6z\pdv{w}{z}$. Její charakteristický systém je $\odv{x}{t} = 2, \odv{y}{t} = 5, \odv{z}{t} = -6z$. Z těchto rovnic dostáváme charakteristiky $\odv*{(5x - 2y)}{t} = 0$ a $\odv*{(3x + \log |z|)}{t} = 0$. + Řešení má tvar $\psi(x, y, z) = F(5x - 2y, 3x + \log z)$. Aplikujeme počáteční podmínku. Platí $\psi(x, 0, x\cos x) = F(5x, 3x + \log |x \cos x|) = 0$. Tomu odpovídá funkce $F(u, v) = 3u - 5v + \log \left(\frac{u}{5}\cos\left(\frac{u}{5}\right)\right)$. Zpětným dosazením dostáváme $0 = 3(5x - 2y) - 5(3x + \log z) + \log(x - \frac{2}{5}y \cos (x - \frac{2}{5}y))$. Z toho máme + $$ z = e^{-\frac{6}{5}y}\left(x - \frac{2}{5}y\cos\left(x - \frac{2}{5}y\right)\right)^{1/5}.$$ +\end{proof} + +Ukážeme si, že řešení nemusí existovat vždy. + +\begin{example} + Dokažte, že neexistuje řešení úlohy $2\pdv{u}{x} + 3\pdv{u}{y} + 8u = 0$ splňující $u(x, \frac{3x - 1}{2}) = e^x)$. +\end{example} + +\begin{proof} + Máme homogenní rovnici $2\pdv{w}{x} + 3\pdv{w}{y} - 8z\pdv{w}{z} = 0$. Její odpovídající charakteristický systém je $\odv{x}{s} = 2, \odv{y}{s} = 3, \odv{z}{s} = -8z$. Z toho snadno dostaneme charakteristiky $\psi_1(x, y, z) = 3x - 2y, \psi_2(x, y, z) = 4x + \log |z|$. + Hledáme řešení ve tvaru $F(3x - 2y, 4x + \log |z|)$, dosadíme podmínku, dostáváme $F(3x - (3x - 1), 4x + \log e^x) = F(1, 5x) = 0$. To může nastat pouze v případě $F \equiv 0$, tedy úloha nemá řešení (kdybychom hledali obecné řešení, dospěli bychom k funkci $u(x, y) = \varphi(3x - 2y)e^{-4x}$). +\end{proof}